Quasiclassical approach to the virial theorem and to the evaluation of the ground-state energy

Using the minimum-energy principle, proofs have been given that the quantum-mechanical virial theorem can be generalized successfully towards non-integrable power-like probe functions. Suitable non-Hermitian momentum operators relying, via dilation operators, on the quasiclassical limit of the usual ones, should then be considered. One proceeds by converting the quantum-mechanical variational problem to a minimization problem of relevant classical Hamiltonians, supplemented by a quantum-mechanically motivated constraint. This constraint expresses the existence of the underlying phase-space quantum. We are then led to identify the ground-state energies with the involved minima of Hamiltonian dispersions. In this way we establish estimates, as well as general properties, concerning the ground-state energies for Schroedinger, Dirac and Klein-Gordon equations with attractive power potentials. Other cases have also been discussed. Stability thresholds have been established correspondingly. (orig.)